PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 94 (108) (2013), 131–140 DOI: 10.2298/PIM1308131M

TWISTED PRODUCT CR-SUBMANIFOLDS IN A LOCALLY CONFORMAL KAEHLER MANIFOLD

Koji Matsumoto and Zerrin Şentürk

Dedicated to Professor Mileva Prvanović on the occasion of her 83rd birthday

Abstract. Recently, we have researched certain twisted product CR-submani-folds in a Kaehler manifold and some inequalities of the second fundamental form of these submanifolds [11].

We consider here two kinds of twisted product CR-submanifolds (the first and the second kind) in a locally conformal Kaehler manifold. In these submanifolds, we give inequalities of the second fundamental form (see Theorems 5.1 and 5.2) and consider the equality case of these.

1. Twisted product manifolds

Let (M_{1}, g_{1}) and (M_{2}, g_{2}) be Riemannian manifolds and M be a (topological) product manifold of M_{1} and M_{2}. We define a Riemannian metric g of M as

g(U, V) = e^{f^{2}} g_{1}(π_{1∗}U, π_{1∗}V) + g_{2}(π_{2∗}U, π_{2∗}V)

for any U, V ∈ T M, where f denotes a positive differentiable function on M, T M is the tangent bundle of M, π_{1} (resp. π_{2}) is a projection operator of M to M_{1} (resp. M_{2}) and π_{1∗} (resp. π_{2∗}) is the differential of π_{1} (resp. π_{2}). Then the manifold M is called a twisted product manifold with an associated (or a warping) function f and we write it M = M_{1} ×_{f} M_{2} [8]. In particular, if the associated function f is in M_{2}, then the manifold M̃ is a warped product [12].

Let M = M_{1}×_{f} M_{2} be a twisted product manifold with the associated function f and let dimM_{1} = n_{1}, dimM_{2} = n_{2} and dimM = n = n_{1} + n_{2}. Moreover, let (x^{1}, x^{2}, …, x^{n_{1}} ), (x^{n_{1} +1} , …, x^{n_{1} +n_{2}} ) be local coordinate systems of M_{1} and M_{2}, respectively. Then (x^{1}, x^{2}, …, x^{n}) is a local coordinate system of M.

2010 Mathematics Subject Classification: 53C40.

Key words and phrases: Twisted product manifold, CR-submanifold, twisted product CRsubmanifold, locally conformal Kaehler manifold.

One of the authors was partially supported by the Serbian Academy.

Using the above local coordinate systems, we can write

(1.1)  (g_{μλ} ) = ( \begin{matrix}g_{ji} & 0 \\ 0 & g_{ba} \end{matrix}) = ( \begin{matrix}e^{f^{2}} g_{1ji} & 0 \\ 0 & g_{2ba} \end{matrix}) ,

where the indices (j, i, …, h), (d, c, …, a) and (ν, μ, …, λ) run over the ranges (1, 2, …, n_{1}), (n_{1} + 1, n_{1} + 2, …, n_{1} + n_{2}) and (1, 2, …, n_{1} + n_{2} = n), respectively.

From (1.1), we have

∂_{k}g_{ji} = e^{f^{2}} {2f^{2}(∂_{k} logf_{1})g_{1ji} + ∂_{k}g_{1ji} },
(1.2)  ∂_{a}g_{ji} = 2f^{2}e^{f^{2}} (∂_{a} logf)g_{1ji} ,
∂_{i}g_{ba} = 0, ∂_{c}g_{ba} = ∂_{c}g_{2ba},

where ∂_{k} = ∂/∂x^{k} and ∂_{a} = ∂/∂x^{a}.

Next, using (1.1) and (1.2), we calculate the Christoffel symbols {_{ν}^{λ}_{μ}} with respect to g_{μλ} which are given by

(1.3)  \{_{ν}^{λ}_{μ}\} = \frac{1}{2} g^{λε} (∂_{ν}g_{εμ} + ∂_{μ}g_{νε} − ∂_{ε}g_{νμ}).

By virtue of (1.2) and (1.3), we obtain

(1.4)

{_{j} ^{h}_{i}} = {_{j} ^{h}_{i}}_{1} + f^{2}{(∂_{j} logf)δ_{i}^{h} + (∂_{i} logf)δ_{j} ^{h} − (∂_{1}^{h} logf)g_{1ji} },
{_{b}^{h}_{i}} = f^{2}(∂_{b}logf)δ_{i}^{h}, {_{b}^{h}_{a}} = 0,
{_{j} ^{a}_{i}} = −f^{2}e^{f^{2}} (∂_{2}^{a} logf)g_{1ji} ,
{_{j} ^{a}_{b}} = 0, {_{c}^{a}_{b}} = {_{c}^{a}_{b}}_{2},

where ∂_{1}^{h} = g_{1}^{lh}∂_{l} (resp. ∂_{2}^{a} = g_{2}^{ea} ∂_{e}) and {_{j} ^{h}_{i}}_{1} (resp. {_{c}^{a}_{b}}_{2}) denotes the Christoffel symbol of g_{1} (resp. g_{2}).

By virtue of (1.4), we get

∇_{Y}X = ∇_{1Y} X + f^{2}{(Y logf)X + (X logf)Y }
− f^{2}g_{1}(Y, X){(∂_{1}^{l} logf)∂_{l} + e^{f^{2}} (∂_{2}^{e} logf)∂_{e}},
∇_{X} Z = ∇_{Z} X = f^{2}(Z logf)X, ∇_{Z} W = ∇_{2Z} W

for any Y, X ∈ T M_{1} and Z, W ∈ T M_{2}, where ∇_{1} (resp. ∇_{2}) denotes the covariant differentiation with respect to g_{1} (resp. g_{2}).

2. Locally conformal Kaehler manifolds

A Hermitian manifold M̃ with structure (J, g̃) is called a locally conformal Kaehler (l.c.K.) manifold if each point x ∈ M̃ has an open neighbourhood U with differentiable function ρ : U → R such that g̃^{∗} = e^{−2ρ} g̃_{|U} is a Kaehlerian metric on U, that is, ∇^{∗}J = 0, where J is the almost complex structure, g̃ is the Hermitian metric, ∇^{∗} is the covariant differentiation with respect to g̃^{∗} and R is a real number space [13]. Then we know [9]

TWISTED PRODUCT CR-SUBMANIFOLDS... 133

Proposition 2.1. A Hermitian manifold M̃ with structure (J, g̃) is l.c.K. if and only if there exists a global 1-form α which is called Lee form satisfying

dα = 0 (α : closed), (2.1)

( ˜∇_{V} J)U = −g̃(α^{♯}, U)JV + g̃(V, U)β^{♯} + g̃(JV, U)α^{♯} − g̃(β^{♯}, U)V (2.2)

for any V, U ∈ M̃, where ˜∇ denotes the covariant differentiation with respect to g̃, α^{♯} is the dual vector field of α, the 1 form β is defined by β(X) = −α(JX), β^{♯} is the dual vector field of β and T M̃ means the tangent bundle of M̃.

3. CR-submanifolds in an l.c.K.-manifold

In general, between a Riemannian manifold (M̃, g̃) and its Riemannian submanifold, we know the Gauss and Weingarten formulas

˜∇_{X} Y = ∇_{X} Y + σ(X, Y),  ˜∇_{X} ξ = −A_{ξ}X + ∇_{X}^{⊥} ξ

for any X, Y ∈ T M and ξ ∈ T^{⊥} M, where σ is the second fundamental form and A_{ξ} is the shape operator with respect to ξ [7]. The second fundamental form σ and the shape operator A are related by g̃(A_{ξ}Y, X) = g̃(σ(Y, X), ξ) for any Y, X ∈ T M and ξ ∈ T^{⊥} M.

A submanifold M in an l.c.K.-manifold M̃ is called a CR-submanifold if there exists a differentiable distribution D : x → D_{x} ⊂ T_{x}M on M satisfying the following conditions

(i) D is holomorphic, i.e., JD_{x} = D_{x} for each x ∈ M and

(ii) the complementary orthogonal distribution D^{⊥} : x → D_{x}^{⊥} ⊂ T_{x}M is totally real, i.e., JD_{x}^{⊥} ⊂ T_{x}^{⊥} M for each x ∈ M, where T_{x}M (resp. T_{x}^{⊥} M) denotes the tangent (resp. normal) vector space at x of M [1, 2, 6, etc.].

If dimD_{x}^{⊥} = 0 (resp. dimD_{x} = 0) for each x ∈ M, then the CR-submanifold is holomorphic (resp. totally real). A CR-submanifold M is said to be anti-holo-morphic if JD_{x}^{⊥} = T_{x}^{⊥} M for any x ∈ M.

In [10], we proved the following

Proposition 3.1. In a CR-submanifold M in an l.c.K.-manifold M̃, we have (i) the distribution D^{⊥} is integrable, (ii) the distribution D is integrable if and only if

(3.1)  g̃(σ(X, JY) − σ(Y, JX) + 2g̃(JX, Y)α^{♯}, JZ) = 0

for any X, Y ∈ D and Z ∈ D^{⊥} .

A CR-submanifold is said to be proper if it is neither holomorphic nor totally real.

In a CR-submanifold M in an l.c.K.-manifold M̃, we know the following formulas [10]

g̃(∇_{U}Z, X) = g̃(JA_{JZ} U, X) + g̃(α^{♯}, Z)g̃(U, X)
+ g̃(U, Z)g̃(α^{♯}, X) − g̃(β^{♯}, Z)g̃(JU, X),
A_{JZ} W = A_{JW} Z + g̃(β^{♯}, Z)W − g̃(β^{♯}, W)Z

for any U ∈ T M, X ∈ D and Z, W ∈ D^{⊥} .

A CR-submanifold is said to be mixed geodesic if the second fundamental form σ satisfies σ(D, D^{⊥} ) = {0} and to be D (resp. D^{⊥} )-geodesic if the second fundamental form σ satisfies σ(D, D) = {0} (resp. σ(D^{⊥} , D^{⊥} ) = {0}).

In a CR-submanifold M of an almost Hermitian manifold M̃, we denote by ν the complementary orthogonal subbundle of JD^{⊥} in the normal bundle T^{⊥} M. Then we have the following direct sum decomposition T^{⊥} M = JD^{⊥} ⊕ ν, JD^{⊥} ⊥ ν.

Remark 3.1. By the definition of ν, a CR-submanifold is anti-holomorphic if ν_{x} = {0} for any x ∈ M.

Definition 3.1. Let M̃ be a Riemannian manifold with a metric tensor g̃. A submanifold M is said to be a twisted product submanifold of M̃ if it satisfies

(i) M is a Riemannian submanifold of M̃,

(ii) M is a twisted product manifold of two submanifolds M_{1} and M_{2} of M̃, (iii) for a certain Riemannian metric g_{1} (resp. g_{2}) of M_{1} (resp. M_{2}),

g(U, V) = e^{f^{2}} g_{1}(π_{1∗}U, π_{1∗}V) + g_{2}(π_{2∗}U, π_{2∗}V)

is an induced metric of g̃ for any U, V ∈ T M̃ and a positive differentiable function f on M, where π_{1} (resp. π_{2}) is the projection operator of M̃ to M_{1} (resp. M_{2}), and π_{1∗} (resp. π_{2∗}) is the differential of π_{1} (resp. π_{2}).

(iv) the submanifolds M_{1} and M_{2} are orthogonal, that is, g̃(X, Z) = 0 for any X ∈ T M_{1} and Z ∈ T M_{2}.

4. Twisted product CR-submanifolds in a locally conformal Kaehler manifold

In this section, we consider a special twisted product submanifold in an l.c.K.- manifold.

Definition 4.1. A submanifold M in an l.c.K.-manifold M̃ is said to be the first (resp. second) kind twisted product CR-submanifold in M̃ if it satisfies

(i) M is a product manifold of a holomorphic submanifold M_{⊤} and a totally real submanifold M_{⊥} ,

(ii) for a certain Riemannian metric tensor g_{1} (resp. g_{2}) on M_{⊤} (resp. M_{⊥} ) and a positive differentiable function f on M,

(4.1)  g(V, U)= e^{f^{2}} g_{1}(π_{∗}V, π_{∗}U) + g_{2}(η_{∗}V, η_{∗}U)
(4.2)  (resp. g(V, U)= g_{1}(π_{∗}V, π_{∗}U) + e^{f^{2}} g_{2}(η_{∗}V, η_{∗}U) )

is a induced metric of g̃, that is, g̃(V, U) = g(i_{∗}V, i_{∗}U), for any V, U ∈ T M, where π (resp. η) is a projection operator of M to M_{⊤} (resp. M_{⊥} ) and i is an identity map of M to M̃.

Then we write the first (resp. second) kind twisted product CR-submanifold M = M_{⊤} ×_{f} M_{⊥} (resp. M = M_{⊥} ×_{f} M_{⊤} ).

Remark 4.1. We write D (resp. D^{⊥} ) instead of T M_{⊤} (resp. T M_{⊥} ).

Remark 4.2. In our submanifold, since the holomorphic distribution D is integrable, we have to assume that the second fundamental form σ satisfies (3.1).

Remark 4.3. About warped product and doubly warped product CR-submani-folds in an l.c.K.-manifold, we can find in [3, 4, 5].

In a CR-submanifold M of an l.c.K.-manifold M̃, let be dimD = 2p, dimD^{⊥} = q, dimM = n, dimν = 2s and dim M̃ = m. Then we know 2p + q = n and 2(p + q + s) = m.

Now we recall an adapted frame on M̃. We take a following local orthonormal frame on M̃,

(i) {e_{1}, e_{2}, …, e_{p}, e^{∗}_{1}, e^{∗}_{2}, …, e^{∗}_{p}} is an orthonormal frame of D,

(ii) {e_{2p+1} , e_{2p+2} , …, e_{2p+q} } is an orthonormal frame of D^{⊥} ,

(iii) {e_{n+q+1} , e_{n+q+2} , …, e_{n+q+s} , e^{∗}_{n+q+1} , e^{∗}_{n+q+2} , …, e^{∗}_{n+q+s} } is an orthonormal frame of ν. Then we know

(a) {e_{1}, …, e_{p}, e^{∗}_{1}, …, e^{∗}_{p}, e_{2p+1} , …, e_{2p+q} } is an orthonormal frame of T M,

(b) {e^{∗}_{2p+1} , …, e^{∗}_{2p+q} , e_{n+q+1} , …, e_{n+q+s} , e^{∗}_{n+q+1} , …, e^{∗}_{n+q+s} } is an orthonormal frame of T^{⊥} M, where e^{∗}_{i} = Je_{i} for i ∈ {1, 2, …, p}, e^{∗}_{2p+a} = Je_{2p+a} for any a ∈ {1, 2, …, q} and e^{∗}_{n+q+α} = Je_{n+q+α} for any α ∈ {1, 2, …, s}. We call such an orthonormal frame {e_{1}, …, e^{∗}_{n+q+s} }, an adapted frame of M̃.

First of all, we consider the first kind twisted product CR-submanifold M in an l.c.K.-manifold M̃. Then, by the definition, the induced metric g on M is defined by (4.1).

Then we have

(4.3)

∇_{Y}X = ∇_{1Y} X + f^{2}{(Y logf)X + (X logf)Y
− f^{2}g_{1}(Y, X){(∂_{1}^{l} logf)∂_{l} + e^{f^{2}} (∂_{2}^{e} logf)∂_{e}},
∇_{X} Z = ∇_{Z} X = f^{2}(Z logf)X, ∇_{Z} W = ∇_{2Z} W

for any Y, X ∈ D and Z, W ∈ D^{⊥} , where ∇_{1} (resp. ∇_{2}) denotes the covariant differentiation with respect to g_{1} (resp. g_{2}).

Proposition 4.1. For a proper first kind twisted product CR-submanifold M = M_{⊤} ×_{f} M_{⊥} in an l.c.K.-manifold M̃, we have

g̃(σ(X, JY), JZ) = g̃(α^{♯}, Z)g̃(X, Y)−g̃(α^{♯}, JZ)g̃(X, JY)−f^{2}(Z logf)g̃(X, Y), (1)

g̃(σ(X, Y), JZ) = g̃(α^{♯}, JZ)g̃(X, Y) and g̃(α^{♯}, Z) = f^{2}(Z logf), (2)

g̃(σ(JX, Z), JW) = −g̃(α^{♯}, X)g̃(Z, W) (3)

for any Y, X ∈ D.

Proof. For any Y, X ∈ D and Z ∈ D^{⊥} , we have from (2.2) and (4.3)

g̃(σ(X, JY), JZ) = g̃( ˜∇_{X} (JY), JZ) = −g̃( ˜∇_{X} (JZ), JY)
= −g̃(( ˜∇_{X} J)Z, JY) − g̃(J ˜∇_{X} Z, JY)
= g̃(α^{♯}, Z)g̃(X, Y) − g̃(α^{♯}, JZ)g̃(X, JY) − g̃(∇_{X} Z, Y).

Using (4.3) and the above equation, we have (1).

In (1), if we put JX instead of X, then we have

g̃(σ(X, Y), JZ) − g̃((α^{♯}, JZ)g̃(X, Y) = {f^{2}(Z logf) − g̃(α^{♯}, Z)}g̃(X, JY).

In the above equation, the left-hand side is symmetric and the right-hand side is skew symmetric with respect to X and Y. So, we have (2).

From (2), (1) is written as

(1′)  g̃(σ(X, JY), JZ) = −g̃(α^{♯}, JZ)g̃(X, JY).

Finally, for (3), we have from (2.2) and (4.3)

g̃(σ(JX, Z), JW) = g̃( ˜∇_{Z} (JX), JW) = g̃(( ˜∇_{Z} J)X, JW) + g̃(J ˜∇_{Z} X, JW)
= g̃(−g̃(α^{♯}, X)JZ + g̃(Z, X)β^{♯} + g̃(JX, Z)α^{♯}
− g̃(β^{♯}, X)Z, JW) + g̃( ˜∇_{Z} X, W) = −g̃(α^{♯}, X)g̃(Z, W)

which means (3). □

By virtue of (2) in the above proposition, we know

Proposition 4.2. There does not exist a proper first kind of twisted product CR-submanifold in an l.c.K.-manifold whose Lee vector field α^{♯} is normal to D^{⊥} .

Proof. By the assumption, we easily know the function f is in M_{⊤} which means our proposition. □

Next, we consider the second kind of twisted product CR-submanifold M = M_{⊥} ×_{f} M_{⊤} in an l.c.K.-manifold M̃. Then, (4.2) means

(g_{μλ} ) = ( \begin{matrix}g_{ba} & 0 \\ 0 & g_{ji} \end{matrix} ) = ( \begin{matrix}e^{f^{2}} g_{2ba} & 0 \\ 0 & g_{1ji} \end{matrix} )
(g^{μλ} ) = ( \begin{matrix}g^{ba} & 0 \\ 0 & g^{ji} \end{matrix} ) = ( \begin{matrix}e^{−f^{2}} g_{2}^{ba} & 0 \\ 0 & g_{1}^{ji} \end{matrix} ) .

In the similar way with a first kind case, we obtain

(4.4)

{_{c}^{a}_{b}} = {_{c}^{a}_{b}}_{2} + f^{2}{(∂_{c} logf)δ_{b}^{a} + (∂_{b}logf)δ_{i}^{a} − (∂_{2}^{a} logf)g_{2cb}},
{_{i}^{a}_{b}} = f^{2}(∂_{i} logf)δ_{b}^{a}, {_{i}^{a}_{h}} = 0,
{_{b}^{h}_{a}} = −f^{2}e^{f^{2}} (∂_{1}^{h} logf)g_{2ba}, {_{b}^{h}_{i}} = 0, {_{j} ^{h}_{i}} = {_{j} ^{h}_{i}}_{1},

Equations (4.4) mean

(4.5)

∇_{Z} W = ∇_{2Z} W + f^{2}{(Z logf)W + (W logf)Z
− f^{2}g_{2}(Z, W){(∂_{2}^{e} logf)∂_{e} + e^{f^{2}} (∂_{1}^{l} logf)∂_{l}},
∇_{Z} X = ∇_{X} Z = f^{2}(X logf)Z, ∇_{Y}X = ∇_{1Y} X

for any Y, X ∈ D, Z, W ∈ D^{⊥} , where ∇_{1} (resp. ∇_{2}) denotes the covariant differentiation with respect to g_{1} (resp. g_{2}).

By virtue of (2.2) and (4.5), we obtain

Proposition 4.3. For the second kind twisted product CR-submanifold M = M_{⊥} ×_{f} M_{⊤} in an l.c.K.-manifold M̃, we have

g̃(σ(Y, JX), JZ) = g̃(α^{♯}, Z)g̃(X, Y) + g̃(α^{♯}, JZ)g̃(X, JY), (1)

g̃(σ(X, Y), JZ) = −g̃(α^{♯}, JZ)g̃(X, Y) and g̃(α^{♯}, Z) = 0, (2)

g̃(σ(JX, Z), JW) = {−g̃(α^{♯}, X) + f^{2}X logf}g̃(Z, W) (3)

for any Y, X ∈ D and Z, W ∈ D^{⊥} .

The proof of Proposition 4.3 is similar to Proposition 4.1. So, we omit it. By virtue of (2) in Proposition 4.3, we know

Proposition 4.4. For the second kind twisted product CR-submanifold M = M_{⊥} ×_{f} M_{⊤} in an l.c.K.-manifold M̃, the Lee vector field α^{♯} is orthogonal to the totally real distribution D^{⊥} , automatically.

5. The length of the second fundamental form

In this section, we consider the length ∥σ∥ of the second fundamental form σ of twisted product CR-submanifolds M = M_{⊤} ×_{f} M_{⊥} and M = M_{⊥} ×_{f} M_{⊤} in an l.c.K.-manifold M̃.

Using the adapted frame, the length ∥σ∥ of the second fundamental form σ is defined as

(5.1)  ∥σ∥^{2} = ∑_{r=n+1}^{m} ∑_{μ,λ=1}^{n} {g̃(σ(e_{λ}, e_{μ}), e_{r})}^{2}.

The equation (5.1) is separated as

∥σ∥^{2} = ∑_{r=n+1}^{n+q} ∑_{μ,λ=1}^{n} {g̃(σ(e_{λ}, e_{μ}), e_{r})}^{2} + ∑_{r=n+q+1}^{m} ∑_{μ,λ=1}^{n} {g̃(σ(e_{λ}, e_{μ}), e_{r})}^{2}

= ∑_{a=1}^{q} ∑_{μ,λ=1}^{n} {g̃(σ(e_{λ}, e_{μ}), Je_{2p+a} )}^{2} + ∑_{r=n+q+1}^{m} ∑_{μ,λ=1}^{n} {g̃(σ(e_{λ}, e_{μ}), e_{r})}^{2}

= ∑_{a=1}^{q} ∑_{j,i=1}^{2p} {g̃(σ(e_{j} , e_{i}), e_{2p+a}^{∗} )}^{2} + 2 ∑_{i=1}^{2p} ∑_{b,a=1}^{q} {g̃(σ(e_{i}, e_{2p+b} ), e_{2p+a}^{∗} )}^{2}

+ ∑_{c,b,a=1}^{q} {g̃(σ(e_{2p+b} , e_{2p+b} ), e_{2p+a}^{∗} )}^{2} + ∑_{r=n+q+1}^{m} ∑_{μ,λ=1}^{n} {g̃(σ(e_{λ}, e_{μ}), e_{r})}^{2},

that is,

∥σ∥^{2} = ∑_{a=1}^{q} ∑_{j,i=1}^{2p} {g̃(σ(e_{j} , e_{i}), e_{2p+a}^{∗} )}^{2} + 2 ∑_{i=1}^{2p} ∑_{b,a=1}^{q} {g̃(σ(e_{i}, e_{2p+b} ), e_{2p+a}^{∗} )}^{2}

+ ∑_{r=n+q+1}^{m} ∑_{μ,λ=1}^{n} {g̃(σ(e_{λ}, e_{μ}), e_{r})}^{2} +∑_{c,b,a=1}^{q} {g̃(σ(e_{2p+c} , e_{2p+b} ), e_{2p+a}^{∗} )}^{2}. (5.2)

Now, assume that our submanifold is the first kind of twisted product CR-sub-manifold M_{⊤} ×_{f} M_{⊥} in an l.c.K.-manifold M̃. Then we have from Proposition 4.1,

g̃(σ(e^{∗}_{i}, e_{2p+a} ), e^{∗}_{2p+b} ) = −g̃(α^{♯}, e_{i})δ_{ba},
g̃(σ(e_{i}, e_{2p+a} ), e_{2p+b}^{∗} ) = g̃(α^{♯}, e_{i}^{∗})δ_{ba},
g̃(σ(e_{i}, e^{∗}_{j}), e_{2p+a} ) = {g̃(α^{♯}, e_{2p+a} ) − f^{2}(e_{2p+a} logf)}δ_{ji} .

for any j, i ∈ {1, 2, …, p} and a, b ∈ {1, 2, …, q}.

Using the above equation, we obtain

∑_{i=1}^{p} ∑_{b,a=1}^{q} g̃(σ(e_{i}^{∗}, e_{2p+a} ), e_{2p+b}^{∗} ) = −q ∑_{i=1}^{p} g̃(α^{♯}, e_{i}),

∑_{i=1}^{p} ∑_{b,a=1}^{q} g̃(σ(e_{i}, e_{2p+a} ), e_{2p+b}^{∗} ) = q ∑_{i=1}^{p} g̃(α^{♯}, e_{j}^{∗}), (5.3)

∑_{j,i=1}^{2p} g̃(σ(e_{i}, e_{j}^{∗}), e_{2p+a}^{∗} ) = 2p{g̃(α^{♯}, e_{2p+a} ) − f^{2}(e_{2p+a} logf)}.

Substituting (5.3) into (5.2), we have

(5.4) ∥σ∥^{2} = 2{p∥α_{D^{⊥}}^{♯} ∥^{2} + q∥α_{D}^{♯}∥^{2}} + 2p ∑_{a=1}^{q} {g̃(α^{♯}, e_{2p+a} ) − f^{2}(e_{2p+a} logf)}^{2}

+ ∑_{c,b,a=1}^{q} {g̃(σ(e_{2p+c} , e_{2p+b} ), e_{2p+a}^{∗} )}^{2} + ∑_{r=n+q+1}^{m} ∑_{μ,λ=1}^{n} {g̃(σ(e_{λ}, e_{μ}), e_{r})}^{2}.

where ∥α_{D^{⊥}}^{♯} ∥ (resp. ∥α_{D}^{♯}∥) denotes the length of α^{♯} in D^{⊥} (resp. D)-part. Hence, we have

(5.5) ∥σ∥^{2} ⩾ 2{p∥α_{D^{⊥}}^{♯} ∥^{2} + q∥α_{D}^{♯}∥^{2}} + 2p ∑_{a=1}^{q} {g̃(α^{♯}, e_{2p+a} ) − f^{2}(e_{2p+a} logf)}^{2}.

Thus we have

Theorem 5.1. In the first kind of twisted product CR-submanifold M = M_{⊤} ×_{f} M_{⊥} in an l.c.K.-manifold M̃, we have (5.4). The equality of (5.5) is satisfied if and only if the second fundamental form σ satisfies σ(D^{⊥} , D^{⊥} ) ⊂ ν and σ(T M, T M) ⊂ JD^{⊥} .

Corollary 5.1. In the first kind of twisted product CR-submanifold M = M_{⊤} ×_{f} M_{⊥} in an l.c.K.-manifold M̃, inequality (5.5) satisfies the equality, then the submanifold M is D^{⊥} -geodesic.

Next, we consider the second kind of twisted product CR-submanifold M = M_{⊥} ×_{f} M_{⊤} . Then we have from Proposition 4.3

g̃(σ(e_{j} , e_{i}), e^{∗}_{2p+a} ) = g̃(α^{♯}, e_{2p+a} )δ_{ji} + g̃(α^{♯}, e^{∗}_{2p+a} )g̃(e_{j} , e^{∗}_{i}),
(5.6)  g̃(σ(e_{j} , e_{i}), e^{∗}_{2p+a} ) = −g̃(α^{♯}, e^{∗}_{2p+a} )δ_{ji} ,

g̃(σ(e^{∗}_{i}, e_{2p+b} ), e^{∗}_{2p+a} ) = {−g̃((α^{♯}, e_{i}) + f^{2}(e_{i} logf)}δ_{ba}

for any j, i ∈ {1, 2, …, 2p} and b, a ∈ {1, 2, …, q}.

Using Proposition 4.4 and equation (5.6), we obtain

∑_{a=1}^{q} ∑_{j,i=1}^{2p} {g̃(σ(e_{j} , e_{i}), e^{∗}_{2p+a} )}^{2} = 2p ∑_{a=1}^{q} {g̃((α^{♯}, e^{∗}_{2p+a} )}^{2}

2 ∑_{i=1}^{p} ∑_{b,a=1}^{q} [g̃(σ(e_{i}, e_{2p+b} ), e^{∗}_{2p+a} )}^{2}

= 2q ∑_{i=1}^{p} [{g̃(α^{♯}, e^{∗}_{i}) − f^{2}(e^{∗}_{i} logf)}^{2} + {g̃(α^{♯}, e_{i}) − f^{2}(e_{i} logf)}^{2}].

Hence the length ∥σ∥ satisfies

∥σ∥^{2} = 2p ∑_{a=1}^{q} {g̃(α^{♯}, e^{∗}_{2p+a} )}^{2}

+ q ∑_{i=1}^{p} [{g̃(α^{♯}, e^{∗}_{i}) − f^{2}(e^{∗}_{i} logf)}^{2} + {g̃(α^{♯}, e_{i}) − f^{2}(e_{i} logf)}^{2}]

+ ∑_{c,b,a=1}^{q} {g̃(σ(e_{2p+c} , e_{2p+b} ), e^{∗}_{2p+a} )}^{2} + ∑_{r=n+q+1}^{m} ∑_{μ,λ=1}^{n} {g̃(σ(e_{μ}, e_{λ}), e_{r})}^{2}

= 2p∥α^{♯}_{JD^{⊥}} ∥^{2} + 2q[∥α_{D}^{♯}∥^{2} − f^{2} ∑_{i=1}^{2p} g̃(α^{♯}, e_{i})(e_{i} logf)

+ f^{4} ∑_{i=1}^{p} {(e_{i} logf)(e_{i}^{∗} logf)}^{2}] + ∑_{c,b,a=1}^{q} {g̃(σ(e_{2p+c} , e_{2p+b} ), e^{∗}_{2p+a} )}^{2}

+ ∑_{r=n+q+1}^{m} ∑_{μ,λ=1}^{n} {g̃(σ(e_{μ}, e_{λ}), e_{r})}^{2},

where α^{♯}_{JD^{⊥}} denotes the JD^{⊥} -component of α^{♯}. Thus we have

(5.7) ∥σ∥^{2} ⩾ 2p ∑_{a=1}^{q} {g̃(α^{♯}, e^{∗}_{2p+a} }^{2}

+ 2q ∑_{i=1}^{p} [{g̃(α^{♯}, e^{∗}_{i}) − f^{2}e^{∗}_{i} logf}^{2} + {g̃(α^{♯}, e_{i}) − f^{2}e_{i} logf}^{2}]

= 2p∥α^{♯}_{jD^{⊥}} ∥^{2} + 2q[∥α_{D}^{♯}∥^{2} − f^{2} ∑_{i=1}^{2p} g̃(α^{♯}, e_{i})(e_{i} logf)

+ f^{4} ∑_{i=1}^{p} {(e_{i} logf)(e_{i}^{∗} logf)}^{2}.

By virtue of (5.7), we obtain

Theorem 5.2. In a second kind twisted product CR-submanifold M = M_{⊥} ×_{f} M_{⊤} in an l.c.K.-manifold M̃, the length ∥σ∥ satisfies inequality (5.7) and equality of (5.7) is satisfied if and only if σ(D^{⊥} , D^{⊥} ) ⊂ ν. and σ(T M, T M) ⊂ JD^{⊥} and then the submanifold M is D^{⊥} -geodesic.

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